Does the CIA have their own jet fighters, or, when they need to carry out a strike, would they need to request it from other branches of the military?
As a civilian agency, the CIA does not officially possess military grade weapons. They may task military authorities- telling a drone pilot where to shoot for example- but the CIA is not military not does it officially hold military hardware.
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Unofficially, it has Jack Ryan - who is way better than Jason Osburne :)
One can never be absolutely sure about a covert intelligence agency, but based upon discussions in military journalism, aircraft journalism, Congressional budget politics, and legal-military journalism (e.g. Lawfare.com) this is my best assessment of the current state of affairs.
The CIA operates armed drones under its own chain of command (as the U.S. has basically admitted).
The CIA has some fixed wing aircraft at its disposal (quoting the Denver Post; see also here) and historically has had some unarmed and armed fixed wing aircraft at its disposal and in its chain of command (for example, B-26 aircraft used in the Bay of Pigs invasion of Cuba and used in Indonesia in the 1950s), but does not have manned armed fixed wing aircraft currently in its chain of command and never had proper fighter jets in its chain of command.
This is corroborated by the fact that the CIA, as of 2013, had a budget of $14.7 billion of which only $2.6 billion was for covert actions including its drone fleet. There simply isn't enough money in its budget to maintain a fleet of manned, fixed wing armed aircraft. Also, fighter pilots need to train regularly and this is a noisy activity visible for tens of miles around that could not easily be conducted covertly outside U.S. military bases where all planes belong to and are operated by the U.S. military branch involved; yet there have been no news reports of CIA fighter jets.
The CIA cannot simply order an air strike absent a pre-existing arrangement with a particular mission, but its intelligence often leads to a strike with U.S. military resources, especially (but not only) special operations forces through a chain of command that includes the President or his senior military advisors, that could include jet fighters (usually dropping bombs, it has been almost half a century the the CIA engaged in manned air to air combat and that was with more or less improvised air resources - one of the few incidents was this one). At times this can be authorized expeditiously (e.g. CIA officers were used as forward observers (i.e. ground troops who call in air strikes) in the U.S. airstrikes by the Air Force following 9-11 in Afghanistan, similar operations were conducted in Laos during the Vietnam War).
It is unclear (to me anyway) if the CIA currently has any armed helicopters in its chain of command. They had helicopters, at least, in Vietnam (one known incident of CIA air to air combat, which was during the Vietnam War, involved a fixed wing enemy aircraft shot down with small arms from a helicopter). An account based upon court documents claims that the CIA purchased Russian military helicopters that it used in the early years of the War in Afghanistan. But, those helicopters may have been discarded as the drone fleet came on line.
Also, as noted in the comments, the U.S. Army does not have any fixed wing fighter aircraft under a "treaty" between the Army and the Air Force reached in 1966. Those resources are housed in the Departments of the Air Force and Navy including the Marine service within the Department of Navy.
It is not really possible to say for sure today, but the CIA has in the past owned and operated its own jet aircraft. It would certainly be reasonable to assume they still do.
However, I don't believe any of these aircraft could be considered "fighters", as they were designed for surveillance, not air superiority. Fighting other aircraft is a primarily military task, so its not really in the CIA's core competency. If they need true military hardware, they'd generally have to beg some off of one of the armed services. If you know anything about big bureaucracies, you'd know this is something they will avoid if at all possible, as they'd likely have to give up some control.
Note that the reverse is most certainly not true. Each of the US military services has its own intelligence agencies. Those by definition have access to heavy military hardware if they should feel they need it.
The key phrase here is 'jet fighters'. They are high performance aircraft whose job is to eliminate other aircraft in a loud, obvious manner. These days, they also operate as tactical bombers. 1000 pound bombs also fall into the category of 'very obvious'.
That is not really the CIA's business. They are in the business of collecting information, and occasionally taking covert action in a 'behind the scenes' manner.
To that end, the CIA funded several aviation projects, most notably the U2 and A12 Oxcart (talk about a clever misnomer - calling the fastest aircraft ever built an ox cart). This later became the SR71, when the USAF adopted the design. Both of those aircraft were designed specifically for collecting information, in a manner previously considered impossible.
A jet fighter just wouldn't be something the CIA would use - too obvious, too traceable to the US. If a situation develops to where interceptors are needed, that situation is no longer covert, and no longer disassociated from the US.